Elliptic boundary value problems of second order in piecewise smooth domains:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Amsterdam [u.a.]
Elsevier
2006
|
| Edition: | 1. ed. |
| Series: | North-Holland mathematical library
69 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 497 - 525 |
| Physical Description: | V, 531 S. |
| ISBN: | 0444521097 9780444521095 |
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| 100 | 1 | |a Borsuk, Mikhail |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Elliptic boundary value problems of second order in piecewise smooth domains |c Mikhail Borsuk ; Vladimir Kondratiev |
| 250 | |a 1. ed. | ||
| 264 | 1 | |a Amsterdam [u.a.] |b Elsevier |c 2006 | |
| 300 | |a V, 531 S. | ||
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| 490 | 1 | |a North-Holland mathematical library |v 69 | |
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| adam_text | 1
Contents
Introduction 7
Chapter 1. Preliminaries 15
1.1. List of symbols 15
1.2. Elementary inequalities 16
1.3. Domains with a conical point 18
1.4. The quasi distance function re and its properties 21
1.5. Function spaces 22
1.5.1. Lebesgue spaces 22
1.5.2. Regularization and Approximation by Smooth Functions . 23
1.6. Holder and Sobolev spaces 26
1.6.1. Notations and definitions 26
1.6.2. Sobolev imbedding theorems 28
1.7. Weighted Sobolev spaces 29
1.8. Spaces of Dini continuous functions 32
1.9. Some functional analysis 36
1.10. The Cauchy problem for a differential inequality 37
1.11. Additional auxiliary results 40
1.11.1. Mean Value Theorem 40
1.11.2. Stampacchia s Lemma 40
1.11.3. Other assertions 41
1.11.4. The distance function 43
1.11.5. Extension Lemma 43
1.11.6. Difference quotients 46
1.12. Notes 48
Chapter 2. Integral inequalities 49
2.1. The classical Hardy inequalities 49
2.2. The Poincare inequality 52
2.3. The Wirtinger inequality: Dirichlet boundary condition... 54
2.4. The Wirtinger inequality: Robin boundary condition 55
2.4.1. The eigenvalue problem 55
2.4.2. The Friedrichs Wirtinger inequality 59
2 Contents
2.5. Hardy Friedrichs Wirtinger type inequalities 60
2.5.1. The Dirichlet boundary condition 60
2.5.2. The Robin boundary condition 68
2.6. Other auxiliary integral inequalities for N = 2 72
2.7. Notes 79
Chapter 3. The Laplace operator 81
3.1. Dini estimates of the generalized Newtonian potential 81
3.2. The equation with constant coefficients. Green s function. 92
3.3. The Laplace operator in weighted Sobolev spaces 95
3.4. Notes 96
Chapter 4. Strong solutions of the Dirichlet problem for linear
equations 97
4.1. The Dirichlet problem in general domains 97
4.2. The Dirichlet problem in a conical domain 100
4.2.1. Estimates in weighted Sobolev spaces 101
4.2.2. The power modulus of continuity 124
4.2.3. Z,p estimates 128
4.2.4. Cx estimates 131
4.2.5. Examples 138
4.2.6. Higher regularity results 140
4.3. Smoothness in a Dini Liapunov region 143
4.4. Unique solvability results 155
4.5. Notes 163
Chapter 5. The Dirichlet problem for elliptic linear divergent
equations in a nonsmooth domain 165
5.1. The best possible Holder exponents for weak solutions 165
5.1.1. Introduction 165
5.1.2. The estimate of the weighted Dirichlet integral 170
5.1.3. Local bound of a weak solution 183
5.1.4. Example 187
5.1.5. Holder continuity of weak solutions 188
5.1.6. Weak solutions of an elliptic inequality 190
5.2. Dini continuity of the first derivatives of weak solutions... 194
5.2.1. Local Dini continuity near a boundary smooth portion 194
5.2.2. Dini continuity near a conical point 198
5.2.3. Global regularity and solvability 209
5.3. Notes 211
Contents 3
Chapter 6. The Dirichlet problem for semilinear equations
in a conical domain 215
6.1. The behavior of strong solutions for nondivergent
equations near a conical point 215
6.1.1. The weighted integral estimates (0 q 1) 216
6.1.2. The estimate of the solution modulus (0 q 1) 220
6.1.3. The estimate of the solution modulus (q 1) 225
6.2. The behavior of weak solutions for divergence equations
near a conical point 232
6.3. Notes 240
Chapter 7. Strong solutions of the Dirichlet problem for
nondivergence quasilinear equations 241
7.1. The Dirichlet problem in smooth domains 241
7.2. The estimate of the Nirenberg type 243
7.2.1. Introduction 243
7.2.2. Formulation of the problem and the main result 244
7.2.3. The Nirenberg estimate 245
7.2.4. The behavior of the solution near a corner point
(weak smoothness) 253
7.2.5. The weighted integral estimate 254
7.2.6. Proof of Theorem 7.7 258
7.3. Estimates near a conical point 263
7.3.1. Introduction 263
7.3.2. The barrier function 264
7.3.3. The weak smoothness of solutions 265
7.3.4. Estimates in weighted spaces 271
7.3.5. Lp and pointwise estimates of the solution
and its gradient 281
7.3.6. Higher regularity results 286
7.4. Solvability results 292
7.5. Notes 297
Chapter 8. Weak solutions of the Dirichlet problem for elliptic
quasilinear equations of divergence form 299
8.1. The Dirichlet problem in general domains 299
8.2. The m—Laplace operator with an absorption term 303
8.2.1. Introduction 303
8.2.2. Singular functions for the m—Laplace operator and
the corresponding eigenvalue problem 304
8.2.3. Eigenvalue problem for m—Laplacian in a bounded
domain on the unit sphere 307
4 Contents
8.2.4. Integral estimates of solutions 314
8.2.5. Estimates of solutions for singular right hand sides 323
8.3. Estimates of weak solutions near a conical point 332
8.4. Integral estimates of second weak derivatives of solutions . 337
8.4.1. Local interior estimates 338
8.4.2. Local estimates near a boundary smooth portion 349
8.4.3. The local estimate near a conical point 354
8.5. Notes 356
Chapter 9. The boundary value problems for elliptic quasilinear
equations with triple degeneration in a domain with
boundary edge 359
9.1. Introduction. Assumptions 359
9.2. A weak comparison principle. The E. Hopf strong
maximum principle 366
9.3. The boundedness of weak solutions 372
9.4. The construction of the barrier function 381
9.4.1. Properties of the function $(w) 383
9.4.2. About solutions of (9.1.3) and (9.1.4) 384
9.4.3. About the solvability of (9.1.3) and (9.1.4) with a0 0 ... 388
9.5. The estimate of weak solutions in a neighborhood
of a boundary edge 394
9.6. Proof of the main theorem 402
9.7. Notes 406
Chapter 10. Sharp estimates of solutions to the Robin boundary
value problem for elliptic non divergence second order
equations in a neighborhood of the conical point 407
10.1. The linear problem 408
10.1.1. Formulation of the main result 408
10.1.2. The Lieberman global and local maximum principle.
The comparison principle 411
10.1.3. The barrier function. The preliminary estimate of
the solution modulus 415
10.1.4. Global integral weighted estimate 420
10.1.5. Local integral weighted estimates 443
10.1.6. The power modulus of continuity at the conical point
for strong solutions 452
10.1.7. Examples 454
10.2. The quasilinear problem 458
10.2.1. Introduction 458
10.2.2. Weak smoothness of the strong solution 461
Contents 5
10.2.3. Integral weighted estimates 469
10.2.4. The power modulus of continuity at the conical point
for strong solutions 488
10.3. Notes 494
Bibliography 497
Index 527
Notation Index 531
|
| adam_txt |
1
Contents
Introduction 7
Chapter 1. Preliminaries 15
1.1. List of symbols 15
1.2. Elementary inequalities 16
1.3. Domains with a conical point 18
1.4. The quasi distance function re and its properties 21
1.5. Function spaces 22
1.5.1. Lebesgue spaces 22
1.5.2. Regularization and Approximation by Smooth Functions . 23
1.6. Holder and Sobolev spaces 26
1.6.1. Notations and definitions 26
1.6.2. Sobolev imbedding theorems 28
1.7. Weighted Sobolev spaces 29
1.8. Spaces of Dini continuous functions 32
1.9. Some functional analysis 36
1.10. The Cauchy problem for a differential inequality 37
1.11. Additional auxiliary results 40
1.11.1. Mean Value Theorem 40
1.11.2. Stampacchia's Lemma 40
1.11.3. Other assertions 41
1.11.4. The distance function 43
1.11.5. Extension Lemma 43
1.11.6. Difference quotients 46
1.12. Notes 48
Chapter 2. Integral inequalities 49
2.1. The classical Hardy inequalities 49
2.2. The Poincare inequality 52
2.3. The Wirtinger inequality: Dirichlet boundary condition. 54
2.4. The Wirtinger inequality: Robin boundary condition 55
2.4.1. The eigenvalue problem 55
2.4.2. The Friedrichs Wirtinger inequality 59
2 Contents
2.5. Hardy Friedrichs Wirtinger type inequalities 60
2.5.1. The Dirichlet boundary condition 60
2.5.2. The Robin boundary condition 68
2.6. Other auxiliary integral inequalities for N = 2 72
2.7. Notes 79
Chapter 3. The Laplace operator 81
3.1. Dini estimates of the generalized Newtonian potential 81
3.2. The equation with constant coefficients. Green's function. 92
3.3. The Laplace operator in weighted Sobolev spaces 95
3.4. Notes 96
Chapter 4. Strong solutions of the Dirichlet problem for linear
equations 97
4.1. The Dirichlet problem in general domains 97
4.2. The Dirichlet problem in a conical domain 100
4.2.1. Estimates in weighted Sobolev spaces 101
4.2.2. The power modulus of continuity 124
4.2.3. Z,p estimates 128
4.2.4. Cx estimates 131
4.2.5. Examples 138
4.2.6. Higher regularity results 140
4.3. Smoothness in a Dini Liapunov region 143
4.4. Unique solvability results 155
4.5. Notes 163
Chapter 5. The Dirichlet problem for elliptic linear divergent
equations in a nonsmooth domain 165
5.1. The best possible Holder exponents for weak solutions 165
5.1.1. Introduction 165
5.1.2. The estimate of the weighted Dirichlet integral 170
5.1.3. Local bound of a weak solution 183
5.1.4. Example 187
5.1.5. Holder continuity of weak solutions 188
5.1.6. Weak solutions of an elliptic inequality 190
5.2. Dini continuity of the first derivatives of weak solutions. 194
5.2.1. Local Dini continuity near a boundary smooth portion 194
5.2.2. Dini continuity near a conical point 198
5.2.3. Global regularity and solvability 209
5.3. Notes 211
Contents 3
Chapter 6. The Dirichlet problem for semilinear equations
in a conical domain 215
6.1. The behavior of strong solutions for nondivergent
equations near a conical point 215
6.1.1. The weighted integral estimates (0 q 1) 216
6.1.2. The estimate of the solution modulus (0 q 1) 220
6.1.3. The estimate of the solution modulus (q 1) 225
6.2. The behavior of weak solutions for divergence equations
near a conical point 232
6.3. Notes 240
Chapter 7. Strong solutions of the Dirichlet problem for
nondivergence quasilinear equations 241
7.1. The Dirichlet problem in smooth domains 241
7.2. The estimate of the Nirenberg type 243
7.2.1. Introduction 243
7.2.2. Formulation of the problem and the main result 244
7.2.3. The Nirenberg estimate 245
7.2.4. The behavior of the solution near a corner point
(weak smoothness) 253
7.2.5. The weighted integral estimate 254
7.2.6. Proof of Theorem 7.7 258
7.3. Estimates near a conical point 263
7.3.1. Introduction 263
7.3.2. The barrier function 264
7.3.3. The weak smoothness of solutions 265
7.3.4. Estimates in weighted spaces 271
7.3.5. Lp and pointwise estimates of the solution
and its gradient 281
7.3.6. Higher regularity results 286
7.4. Solvability results 292
7.5. Notes 297
Chapter 8. Weak solutions of the Dirichlet problem for elliptic
quasilinear equations of divergence form 299
8.1. The Dirichlet problem in general domains 299
8.2. The m—Laplace operator with an absorption term 303
8.2.1. Introduction 303
8.2.2. Singular functions for the m—Laplace operator and
the corresponding eigenvalue problem 304
8.2.3. Eigenvalue problem for m—Laplacian in a bounded
domain on the unit sphere 307
4 Contents
8.2.4. Integral estimates of solutions 314
8.2.5. Estimates of solutions for singular right hand sides 323
8.3. Estimates of weak solutions near a conical point 332
8.4. Integral estimates of second weak derivatives of solutions . 337
8.4.1. Local interior estimates 338
8.4.2. Local estimates near a boundary smooth portion 349
8.4.3. The local estimate near a conical point 354
8.5. Notes 356
Chapter 9. The boundary value problems for elliptic quasilinear
equations with triple degeneration in a domain with
boundary edge 359
9.1. Introduction. Assumptions 359
9.2. A weak comparison principle. The E. Hopf strong
maximum principle 366
9.3. The boundedness of weak solutions 372
9.4. The construction of the barrier function 381
9.4.1. Properties of the function $(w) 383
9.4.2. About solutions of (9.1.3) and (9.1.4) 384
9.4.3. About the solvability of (9.1.3) and (9.1.4) with a0 0 . 388
9.5. The estimate of weak solutions in a neighborhood
of a boundary edge 394
9.6. Proof of the main theorem 402
9.7. Notes 406
Chapter 10. Sharp estimates of solutions to the Robin boundary
value problem for elliptic non divergence second order
equations in a neighborhood of the conical point 407
10.1. The linear problem 408
10.1.1. Formulation of the main result 408
10.1.2. The Lieberman global and local maximum principle.
The comparison principle 411
10.1.3. The barrier function. The preliminary estimate of
the solution modulus 415
10.1.4. Global integral weighted estimate 420
10.1.5. Local integral weighted estimates 443
10.1.6. The power modulus of continuity at the conical point
for strong solutions 452
10.1.7. Examples 454
10.2. The quasilinear problem 458
10.2.1. Introduction 458
10.2.2. Weak smoothness of the strong solution 461
Contents 5
10.2.3. Integral weighted estimates 469
10.2.4. The power modulus of continuity at the conical point
for strong solutions 488
10.3. Notes 494
Bibliography 497
Index 527
Notation Index 531 |
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| id | DE-604.BV022121968 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T16:16:18Z |
| indexdate | 2024-07-09T20:50:57Z |
| institution | BVB |
| isbn | 0444521097 9780444521095 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015336679 |
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| physical | V, 531 S. |
| publishDate | 2006 |
| publishDateSearch | 2006 |
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| publisher | Elsevier |
| record_format | marc |
| series | North-Holland mathematical library |
| series2 | North-Holland mathematical library |
| spelling | Borsuk, Mikhail Verfasser aut Elliptic boundary value problems of second order in piecewise smooth domains Mikhail Borsuk ; Vladimir Kondratiev 1. ed. Amsterdam [u.a.] Elsevier 2006 V, 531 S. txt rdacontent n rdamedia nc rdacarrier North-Holland mathematical library 69 Literaturverz. S. 497 - 525 Boundary value problems Differential equations, Elliptic Elliptisches Randwertproblem (DE-588)4193399-0 gnd rswk-swf Elliptisches Randwertproblem (DE-588)4193399-0 s DE-604 Kondrat'ev, Vladimir Verfasser aut North-Holland mathematical library 69 (DE-604)BV000005206 69 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015336679&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Borsuk, Mikhail Kondrat'ev, Vladimir Elliptic boundary value problems of second order in piecewise smooth domains North-Holland mathematical library Boundary value problems Differential equations, Elliptic Elliptisches Randwertproblem (DE-588)4193399-0 gnd |
| subject_GND | (DE-588)4193399-0 |
| title | Elliptic boundary value problems of second order in piecewise smooth domains |
| title_auth | Elliptic boundary value problems of second order in piecewise smooth domains |
| title_exact_search | Elliptic boundary value problems of second order in piecewise smooth domains |
| title_exact_search_txtP | Elliptic boundary value problems of second order in piecewise smooth domains |
| title_full | Elliptic boundary value problems of second order in piecewise smooth domains Mikhail Borsuk ; Vladimir Kondratiev |
| title_fullStr | Elliptic boundary value problems of second order in piecewise smooth domains Mikhail Borsuk ; Vladimir Kondratiev |
| title_full_unstemmed | Elliptic boundary value problems of second order in piecewise smooth domains Mikhail Borsuk ; Vladimir Kondratiev |
| title_short | Elliptic boundary value problems of second order in piecewise smooth domains |
| title_sort | elliptic boundary value problems of second order in piecewise smooth domains |
| topic | Boundary value problems Differential equations, Elliptic Elliptisches Randwertproblem (DE-588)4193399-0 gnd |
| topic_facet | Boundary value problems Differential equations, Elliptic Elliptisches Randwertproblem |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015336679&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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